Question
The Tangent Ratio Quick Check 4 of 54 of 5 Items Question Xavier is looking up at a building that is 123 meters away from him, at an angle of elevation of 52° . What is the building’s height above Xavier’s eye level, x, and the distance between Xavier and the top of the building, y? (1 point) Responses x=157.43 m and y=199.79 m x equals 157.43 m and y equals 199.79 m x=96.1 m and y=156.09 m x equals 96.1 m and y equals 156.09 m x=199.79 m and y=157.43 m x equals 199.79 m and y equals 157.43 m x=157.43 m and y=98.26 m x equals 157.43 m and y equals m Skip to navigation page 8 of 8
Answers
GPT-4o mini
To find the height of the building above Xavier's eye level (x) and the distance between Xavier and the top of the building (y), we can use some trigonometric principles.
1. **Finding the Height Above Eye Level (x):**
- We know that the angle of elevation is \(52^\circ\), and the horizontal distance to the building is \(123\) meters.
- We can use the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Hence,
\[
\tan(52^\circ) = \frac{x}{123}
\]
- Rearranging to find x gives:
\[
x = 123 \cdot \tan(52^\circ)
\]
- Using a calculator,
\[
x \approx 123 \cdot 1.2799 \approx 157.43 \text{ meters}
\]
2. **Finding the Distance to the Top of the Building (y):**
- To find the distance (y) from Xavier to the top of the building, we can use the cosine function in combination with the Pythagorean theorem.
- We can find y using:
\[
y = \frac{x}{\sin(52^\circ)}
\]
- First, we need to find the slant distance using the Pythagorean theorem:
\[
y = \sqrt{(123)^2 + (x)^2}
\]
- Substitute \(x\):
\[
y = \sqrt{(123)^2 + (157.43)^2}
\]
- Calculate:
\[
y \approx \sqrt{15129 + 24836.0649} \approx \sqrt{39965.0649} \approx 199.91 \text{ meters}
\]
In summary:
- \( x \approx 157.43 \text{ m} \)
- \( y \approx 199.91 \text{ m} \)
Based on your options:
- \( x = 157.43 \text{ m} \) and \( y = 199.79 \text{ m} \) would be the closest accurate response.
**Final Answers:**
\( x = 157.43 \: m \) and \( y = 199.79 \: m \)
1. **Finding the Height Above Eye Level (x):**
- We know that the angle of elevation is \(52^\circ\), and the horizontal distance to the building is \(123\) meters.
- We can use the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Hence,
\[
\tan(52^\circ) = \frac{x}{123}
\]
- Rearranging to find x gives:
\[
x = 123 \cdot \tan(52^\circ)
\]
- Using a calculator,
\[
x \approx 123 \cdot 1.2799 \approx 157.43 \text{ meters}
\]
2. **Finding the Distance to the Top of the Building (y):**
- To find the distance (y) from Xavier to the top of the building, we can use the cosine function in combination with the Pythagorean theorem.
- We can find y using:
\[
y = \frac{x}{\sin(52^\circ)}
\]
- First, we need to find the slant distance using the Pythagorean theorem:
\[
y = \sqrt{(123)^2 + (x)^2}
\]
- Substitute \(x\):
\[
y = \sqrt{(123)^2 + (157.43)^2}
\]
- Calculate:
\[
y \approx \sqrt{15129 + 24836.0649} \approx \sqrt{39965.0649} \approx 199.91 \text{ meters}
\]
In summary:
- \( x \approx 157.43 \text{ m} \)
- \( y \approx 199.91 \text{ m} \)
Based on your options:
- \( x = 157.43 \text{ m} \) and \( y = 199.79 \text{ m} \) would be the closest accurate response.
**Final Answers:**
\( x = 157.43 \: m \) and \( y = 199.79 \: m \)